Can quran be recited in the state of periods?
Where we are not touching Quran and only reading it?

Yes, according to Ayatullah Seestani, you can read the Quran except for the those verses which include obligatory Sajdas, although it is not allowed to touch the verses in such a state.
According to some jurists, when one cleans themselves during the prayer time and recites the Holy Quran will earn the same reward as if one is praying.