Short answer
No. Elohim is the usual Hebrew word for God (﷾), and it is plural in form. But when it means the one God (﷾), the Bible almost always gives it singular verbs. The same word is used of one pagan god, Chemosh, worshipped by Israel's neighbours, and even of Moses. Hebrew grammar calls this a plural of majesty, a plural that shows greatness, not number. In "Let us make man", one God (﷾) is speaking. He is either thinking aloud or, as Jewish readers say, speaking to His angels. The very next verse says "God created man in his own image", with a singular verb. The Jews, whose scripture this is, never read a Trinity into these verses. Islam's answer is that God (﷾) is One. In the Qur'an too, the One God (﷾) says "We", to show His greatness.
Full explanation
The verses Christians point to
The Trinity is the Christian teaching that the one God is three persons: Father, Son and Holy Spirit. Some Christians see a hint of this in the plural word Elohim. They also see it in the places where God (﷾) says "us". These are the creation of man, Adam's sin, the Tower of Babel, and a vision of God's (﷾) throne seen by the prophet Isaiah.
“And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth.”
“And the Lord God said, Behold, the man is become as one of us, to know good and evil: and now, lest he put forth his hand, and take also of the tree of life, and eat, and live for ever:”
“Go to, let us go down, and there confound their language, that they may not understand one another’s speech.”
“Also I heard the voice of the Lord, saying, Whom shall I send, and who will go for us? Then said I, Here am I; send me.”
Elohim takes singular verbs
Start with the word itself. Elohim is plural in form. But when it means the God (﷾) of Israel, the verbs that go with it are almost always singular. In the Bible's first verse, "created" (bara in Hebrew) is singular. Gesenius' Hebrew Grammar, the standard reference book on biblical Hebrew (English edition, 1910), calls this a plural of majesty. That is a plural that expresses greatness, not number.
“In the beginning God created the heaven and the earth.”
The standard reference grammar of biblical Hebrew. It classes Elohim, used of the one God, as a plural of majesty or excellence (§124g), and notes that in that sense it is regularly construed with singular verbs and adjectives (§132h, §145h). It says God's "us" in Genesis 1:26, 11:7 and Isaiah 6:8 is not a plural of majesty. It is best explained as a plural of self-deliberation, though at least in Isaiah 6:8 (compare Genesis 3:22) it includes the attendant angels (§124g, note).
The same word for Chemosh and for Moses
The Bible uses the same word for a single pagan god, Chemosh, whom Israel's neighbours worshipped. It also uses it for Moses, whom God (﷾) made "a god to Pharaoh". Nobody thinks Chemosh or Moses was three persons. So a plural word does not mean three persons.
“Wilt not thou possess that which Chemosh thy god giveth thee to possess? So whomsoever the Lord our God shall drive out from before us, them will we possess.”
“And the Lord said unto Moses, See, I have made thee a god to Pharaoh: and Aaron thy brother shall be thy prophet.”
The next verse answers the first
Now turn to the "us" verses. Straight after "Let us make man", Genesis says God (﷾) created man in his own image. The verb is singular, and so is the word "his". The text never says "they created", and it never says how many the "us" are. The prophet Isaiah adds that God (﷾) stretched out the heavens alone and spread out the earth by Himself. Whoever the "us" is, it is not a partner in creation.
“So God created man in his own image, in the image of God created he him; male and female created he them.”
“Thus saith the Lord, thy redeemer, and he that formed thee from the womb, I am the Lord that maketh all things; that stretcheth forth the heavens alone; that spreadeth abroad the earth by myself;”
One speaker can say "we"
One speaker can use "we" for himself. The book of Ezra is partly in Aramaic, a language related to Hebrew. There a Persian king writes that the letter sent "unto us" was read "before me". Many readers take this as a king speaking of himself in the plural. In Isaiah's vision, one voice says both "Whom shall I send" and "who will go for us". The Hebrew grammar book explains God's (﷾) "Let us" as one speaker thinking aloud. In Isaiah's vision it explains "us" as God (﷾) including the angels who attend Him. Neither reading counts persons in God (﷾).
“The letter which ye sent unto us hath been plainly read before me.”
“Also I heard the voice of the Lord, saying, Whom shall I send, and who will go for us? Then said I, Here am I; send me.”
The standard reference grammar of biblical Hebrew. It classes Elohim, used of the one God, as a plural of majesty or excellence (§124g), and notes that in that sense it is regularly construed with singular verbs and adjectives (§132h, §145h). It says God's "us" in Genesis 1:26, 11:7 and Isaiah 6:8 is not a plural of majesty. It is best explained as a plural of self-deliberation, though at least in Isaiah 6:8 (compare Genesis 3:22) it includes the attendant angels (§124g, note).
How Jewish readers take it
These verses belong to the Jewish scripture. Jews have read them for thousands of years without becoming Trinitarian. Rashi, the classic Jewish commentator (11th century), reads "Let us" as God (﷾) consulting His angels, to teach humility. He points out that the next verse says "He created", not "they created". The creed of Israel stays the same: the Lord is one.
The classic Jewish commentary reads "Let us make man" as God consulting His heavenly court of angels out of humility, though they did not help Him create. In the same comment it points to the next verse, which says "He created", not "they created", as the answer to anyone who would find more than one creator here.
“Hear, O Israel: The Lord our God is one Lord:”
The Qur'an's "We"
In the Qur'an God often says "We". It speaks of the Message that We sent down, and of the heavens and the earth that We created. The same Qur'an says God (﷾) is One, with none like Him. Muslims have never read that "We" as more than one God (﷾). In the same way, a plural pronoun does not make the God (﷾) of Genesis more than one.
“We have, without doubt, sent down the Message; and We will assuredly guard it (from corruption).”
“We created the heavens and the earth and all between them in Six Days, nor did any sense of weariness touch Us.”
“Say: He is Allah, the One and Only; Allah, the Eternal, Absolute; He begetteth not, nor is He begotten; And there is none like unto Him.”
Verse by verse
Genesis 1:26 — Torah, Bereshit, chapter 1, verse 26
“And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth.”
Cited for
"Let us make man in our image" is God the Father speaking with the Son and the Spirit.
The response
The next verse reports what happened: God (﷾) created man in his own image, with a singular verb. The text never says they created.
Where it lands — The "us" does not count persons. The creating is told in the singular, and Jewish readers take the "us" as God (﷾) with His angels.
Genesis 3:22 — Torah, Bereshit, chapter 3, verse 22
“And the Lord God said, Behold, the man is become as one of us, to know good and evil: and now, lest he put forth his hand, and take also of the tree of life, and eat, and live for ever:”
Cited for
"The man is become as one of us" shows more than one divine person.
The response
The next verse again uses the singular. The Lord God sent him out of the garden. The same pattern, a plural "us" followed by a singular action, runs through the story.
Where it lands — A plural "us" followed by a singular act is a way of speaking, as at Babel. It is not a count of persons.
Genesis 11:7 — Torah, Bereshit, chapter 11, verse 7
“Go to, let us go down, and there confound their language, that they may not understand one another’s speech.”
Cited for
"Let us go down" at the Tower of Babel is the persons of the Trinity acting together.
The response
The next verse says: "So the Lord scattered them abroad". One Lord does what "us" announced.
Where it lands — The plural announces, and the singular acts. This is a way of speaking, not a count of persons.
Perspectives
Trinitarian reading
The plural Elohim and God's "us" in Genesis leave room for, or hint at, more than one person within the one God. Christians say the New Testament reveals this more fully.
In Genesis 1:26 God says "Let us make man in our image". Some Christians read this as God speaking among the persons of the Trinity.
Rebuttal
Christians agree God (﷾) is one being, so the singular verb alone does not settle it. But nothing in the verse says three. The standard Hebrew grammar reads the "us" as one speaker thinking aloud, and Jewish readers take it as God (﷾) with His angels.
Jewish
The plural is majesty, or God speaking to His heavenly court. The next verse says He created alone, and God is one.
Rashi reads "Let us" as God consulting the angels, and stresses that the next verse uses the singular "He created".
Rebuttal
Islam agrees with Rashi's main point: the next verse says "He created", not "they created". Whoever the "us" includes, only one Creator acts. Isaiah says God (﷾) stretched out the heavens alone.
The Torah's creed declares that the Lord is one.
Rebuttal
Islam agrees: the Qur'an calls God "the One and Only", with none like Him. Jesus called this creed the first commandment. When a scribe replied "there is none other but he", Jesus said he was not far from the kingdom of God (﷾).
The Islamic perspective
God (﷾) is One. The plural Elohim is a plural of majesty. The "us" is one God speaking, thinking aloud or addressing His angels. In the Qur'an too the One God (﷾) says "We".
When Elohim means the God (﷾) of Israel, it almost always takes singular verbs. Hebrew grammar calls this a plural of majesty.
The Bible uses Elohim of the single pagan god Chemosh and of Moses.
Genesis 1:27 follows "Let us make man" with a singular verb and "his own image".
A Persian king in Ezra writes that the letter sent "unto us" was read "before me". This is widely read as a royal plural.
The Qur'an has the One God (﷾) speak as "We" while teaching strict oneness.