I have seen this quote in multiple books by the Turkish poet, Ahmet Necip Fazıl Kısakürek, who actually was a student of Henri Bergson.
However, I am not aware if this quote appears in one of Bergson's works.
The book, Mümin - Kâfir (The Believer and Disbeliever) has following dialogue:
Believer - As a matter of fact, a Western philosopher who has
destroyed the rationalist system before him, to his opponents saying
“you might have destroyed the rationalist system, but your methods
used reason. What can you say about that?” replied by saying
“Therefore, the supreme and ultimate activity of the mind is
destroying itself with its own methods…” Do you know this philosopher?
Disbeliever – Yes… The famous Bergson… One of the poet philosophers…
The original text [in Turkish] can be found here.
If anyone can point out any works this quote has been featured in, it would be great.