Short answer
No. The verse tells what a traveller saw, not where the sun goes. Qur'an 18:86 says that Dhul-Qarnayn, a ruler named in the Qur'an, travelled west until he reached the setting of the sun. There "he found it" setting in a spring of murky water. Anyone who stands on a western shore at evening sees the sun sink into the water. Four verses later, the same wording says he came to the rising of the sun, in the east. That plainly does not mean the sun comes up out of the ground. Elsewhere the Qur'an says the sun swims along in its own orbit. The commentator Ibn Kathir read the verse this way in the fourteenth century. Islam's answer is that the Qur'an describes what a man saw, and never teaches that the sun drops into a pool.
Full explanation
What the verse says
The Qur'an tells the story of Dhul-Qarnayn, a ruler given power on earth, who made journeys to the west and to the east. At the far west, at the place where the sun sets, he found it setting in dark water. At the far east he found it rising on a people with no shelter from it. Both verses use the same verb, wajada, which means "he found it". The word for the water is read in two recognised ways, hami'a (muddy) and hamiya (hot). Either way, it describes the water he saw.
“They ask thee concerning Zul-qarnain. Say, "I will rehearse to you something of his story."”
“Until, when he reached the setting of the sun, he found it set in a spring of murky water: Near it he found a People: We said: "O Zul-qarnain! (thou hast authority,) either to punish them, or to treat them with kindness."”
“Until, when he came to the rising of the sun, he found it rising on a people for whom We had provided no covering protection against the sun.”
"He found it": a traveller's view
The verse does not say the sun sets in a spring. It says he found it setting there. That is what his eyes saw when he reached the western sea. The pair of verses makes this plain. He also came to "the rising of the sun". That plainly does not mean a spot on earth where the sun climbs out of the ground. Both verses describe the sun as it looked to a man standing at the edge of the land. In the same verse he also "found" a people, and they were real. So was the sunset he found: the sun going down into dark water, as it looks from any shore. We still speak the same way. A newspaper that prints the time of sunset is not teaching that the sun goes round the earth.
“Until, when he reached the setting of the sun, he found it set in a spring of murky water: Near it he found a People: We said: "O Zul-qarnain! (thou hast authority,) either to punish them, or to treat them with kindness."”
“Until, when he came to the rising of the sun, he found it rising on a people for whom We had provided no covering protection against the sun.”
Where the Qur'an puts the sun
When the Qur'an speaks about the sun itself, and not about what someone saw, it places the sun in an orbit. It says the sun and the moon each swim along in their own course. It also says the sun is not permitted to catch up the moon. A book that says this cannot also be teaching that the sun drops into a pool on earth.
“It is He Who created the Night and the Day, and the sun and the moon: all (the celestial bodies) swim along, each in its rounded course.”
“It is not permitted to the Sun to catch up the Moon, nor can the Night outstrip the Day: Each (just) swims along in (its own) orbit (according to Law).”
How the commentators read it
Muslim scholars did not need modern astronomy to read the verse correctly. Ibn Kathir was a Muslim commentator of the fourteenth century. He explained that Dhul-Qarnayn saw the sun as though it were setting in the ocean, as anyone on a shore does. He added that the sun never leaves its course.
Explains that Dhul-Qarnayn saw the sun as if it were setting into the ocean, as anyone who goes to the shore sees it, while the sun itself never leaves the course in which it is fixed.
“Until, when he reached the setting of the sun, he found it set in a spring of murky water: Near it he found a People: We said: "O Zul-qarnain! (thou hast authority,) either to punish them, or to treat them with kindness."”
What kind of objection this is
The Qur'an invites its readers to look for contradictions in it. This objection is really about what the verse describes. Is a traveller's view a claim about astronomy? That is different from two statements in one book that cannot both be true, such as two numbers for the same event.
“Do they not consider the Qur'an (with care)? Had it been from other Than Allah, they would surely have found therein Much discrepancy.”
“Until, when he reached the setting of the sun, he found it set in a spring of murky water: Near it he found a People: We said: "O Zul-qarnain! (thou hast authority,) either to punish them, or to treat them with kindness."”
What critics say, and the answer
Qur'an 18:86 — Surat al-Kahf (The Cave), verse 86
“Until, when he reached the setting of the sun, he found it set in a spring of murky water: Near it he found a People: We said: "O Zul-qarnain! (thou hast authority,) either to punish them, or to treat them with kindness."”
Cited for
Critics say the verse teaches that the sun sets in a spring of mud somewhere on earth, which is false.
The response
The verse says he found it setting there, which is what anyone sees at the western sea. He also "found" a people there, and they were real. So was the sunset: the sun going down into dark water, as it looks from any shore. The matching verse says he found it rising on a people in the far east. That plainly does not mean the sun climbs out of the ground there. And the Qur'an places the sun itself in an orbit.
Where it lands — The verse describes a sunset as a traveller saw it. It is not a claim about where the sun goes.
Perspectives
The Islamic perspective
Qur'an 18:86 describes the sunset as Dhul-Qarnayn saw it at the western sea. It makes no claim about where the sun goes, and the Qur'an elsewhere places the sun in its own orbit.
The verse says Dhul-Qarnayn "found" the sun setting in murky water, which reports what he saw.
Four verses later the same wording says he found the sun rising on a people in the far east. That plainly does not mean the sun comes up out of the ground there, so both verses describe his view.
When the Qur'an speaks of the sun itself, it says the sun swims along in its own orbit.
Ibn Kathir explained the verse as the sun seen setting into the ocean from the shore. The sun itself never leaves its course.
The Qur'an teaches that it would contain many contradictions if it were from other than Allah (﷾). This objection does not show one.