Where is the sole survivor "quarters" in the Institute that Father provides?

Solution 1:

Father will tell you to go to your quarters after you come back from the Libertalia during Synth Retention. This should be your current quest objective, and you need to do it for the quest to complete.

From Nukapedia:

On returning to the Institute there will be another discussion with Father, he'll provide quarters. Upon entering them the mission is complete and The Battle of Bunker Hill begins.

Quarters Location

I don't recall if there's a quest marker to your quarters.

There are four four-story towers spread out radially around the institute. Each tower has a room in the base before the stairs, as well as a symbol over the archway to this room.

The four rooms are:

• A shop, with a green bag icon
• A cafeteria, with a yellow plate, fork, and knife icon
• A clinic, with a red cross icon
• And some sort of waiting area with benches and a blue clock icon.

Father's quarters are in the tower above the shop. Your quarters are on the third floor of the next tower clockwise, which has the clock icon. From the stairway, there is a hallway with a living quarters on each side. The hallway leads to a platform connecting to the central elevator.

Your quarters are on the left side of the hallway. You can tell because you can sleep in the bed and become well rested.

Right after Synth Retention, X6-88 should be standing outside your quarters.

Nukapedia does mention a possible related bug:

X6-88 may not appear next to your quarters at the Institute, preventing the final objective from being completed. To fix it, fast travel to Libertalia and speak to X6-88. Unfortunately on Xbox One and PS4 this method doesn't always work. To further fix this, he will mostly at all times be located at a small boat directly between Libertalia and Nordhagen Beach. Simply locate him and being shooting him. After he takes some damage, the game will prompt as the quest completed. If you desire to acquire him later on, do not finish him off and instead evade him.

Solution 2:

Blue Icon tower, second floor, left room.

At least there is a bed where you can sleep, not 100% certain that this is the room provided by father (feel free to comment if you know better).