Yes of course one can participate in both All India Quota Counselling and State Quota Counselling as both the dates never clash with each other .

It is done so that the students who is getting top colleges through any of the Counselling can take part in the Counselling process.

There are three Rounds in NEET UG Counselling which are as follows :

1. First Round Counselling
2. Second Round Counselling
3. Mop Up Round Counselling

The State Quota Counselling and All India Counselling process happen in following way :

First the students have to register for All India Counselling which is done by Medical Counselling Committee . After first Round of All India Quota Counselling the Round first for State Quota Counselling starts and after that 2nd Round for All India and then State Quota Counselling for second Round Counselling process starts and same way the Mop Up Round Counselling.

But there's a catch. Once you are alloted a seat through AIQ in Second Round, you have to take admission in it or your security money will be forfeited. And if you take admission through AIQ in Second Round then you won't be allowed to leave that seat.

So be careful and plan properly before making a decision.