How long after a FUE hair transplant can scalp micropigmentation be done?

It depends on the doctor's comfort level.  Usually Scalp MicroPigmentaion can be done weeks to months after FUE.  

SMP, or scalp micropigmentation, is a painless procedure that uses highly specialized equipment and unique techniques to carefully and expertly inject all-natural, color-tailored pigments into the scalp creating a completely natural look of the patient’s hairline. A typical procedure involves an initial consultation and two pigmentation sessions to achieve the desired results. You can wash your head the fourth day after the treatment and, during that time, avoid any strenuous activity that will prompt undesired sweating. Once the procedure is complete, the results of a Scalp Micropigmentation treatment can last up to five years.
At the Foundation for Hair Restoration in NYC, I work with the world-renowned hair expert Dr.Jeffrey Epstein and his team to create beautiful, natural results for any hair type.
SMP can safely be done about 3 months after a transplant. 
Consult with an experienced hair restoration surgeon for a complete range of options that are appropriate for your situation.

Scalp micro pigmentation or SMP is a perfect solution to not only cover the scars after and follicular unit extraction (FUE) hair transplant, but it can also camouflage any remaining bald spots on the scalp that have not been covered by the dispersed follicles in the recipient site.
This procedure helps hit two birds with one stone and is widely sought after by many patients both prior and post hair transplantation. Now coming to the real question; When can a patient become eligible to undergo scalp micro pigmentation after a hair transplant?
A hair transplant is a meticulous procedure that does put the scalp under duress by shifting and distributing follicles from one region to another. Quite similar to re-sowing seeds from one part of land into another. This is why after the procedure, the patient will feel tenderness on their scalp.
Scalp micro pigmentation does something similar but in a much lower degree. The needle used to deposit the ink can make the scalp irritated and can affect either donor or recipient sites of the scalp if done too early on after the procedure.
The most sound bet is to let the scalp heal from its tenderness and allow the new follicles to settle in place and begin their growth before proceeding with scalp micro pigmentation. Therefore, a patient becomes eligible for scalp micro pigmentation after four months of an FUE hair transplant.

After FUE you ideally need to wait for the hair growth to begin to mature. The idea of scalp pigmentation is to create the look of greater fullness to your hair. For this to be made effective it is best to see the original hair density you have, so allowing a few months for your FUE to grow. This can help to judge if some areas require greater attention to pigmentation to others and create an even look of fullness.  

I typically recommend waiting at least 6 months post hair transplant before doing any SMP in either the donor or recipients sites. 

The SMP process can be done between 3-6 months after a hair transplant based upon your individual circumstances. You need to discuss your goals for this process and if you intend to keep your hair long or shave your head for the procedure. See Web Reference below for pictures and explantations.

Depending on what you are looking to accomplish would help to determine the timing. If you are looking for the SMP to fill in between the grafts or conceal the small circular scars from the FUE harvest, I would think you should wait at least 9 months for grafts to grow out and donor scarring to become more mature.