Do we say:
Can you please write up a testimonial about me on LinkedIn?
or
Can you please write up a testimonial of me in LinkedIn?
Can you write (up) a testimonial about me on LinkedIn? is perfectly acceptable.
As a native speaker, I would personally say:
Would you mind posting a testimonial about me on LinkedIn?
It depends!
You write testimonial for someone.
Could you please write a testimonial for me on my LinkedIn profile?
Someone can also give a testimonial. The example from Collins:
If I told them the truth, they had no reason to believe me; Hebburn wasn't going to give me a glowing testimonial.
For this particular case, it can work without preposition as well:
Could you please write a testimonial on my LinkedIn profile? - Since you are defining 'your' LinkedIn profile, it's undrestood that the testimonial is for you and not anyone else.
You can also have testimonial about something:
Could you please write a testimonial about my work with your firm?