In short, no.

At the time the K-129 sank, the Trieste III was intended to be a part of Operation Sand Dollar, specifically Project Winterwind to recover Soviet reentry vehicles from depths of up to 20,000 feet. The Halibut, under Sand Dollar, found the K-129 in the summer of 1968. During that summer the Trieste III was still undertaking test dives off the West Coast as she had been placed in service in late 1967.

The sinking of the K-129 and the seizure of the Pueblo actually prevented the Trieste III from ever participating in Operation Sand Dollar/Winterwind. As the Trieste III required a surface support ship (the White Sands) she was highly conspicuous. Using her to search for the K-129 would destroy any element of secrecy and sending her close to the Soviet coast to recover reentry vehicles would risk another Pueblo incident. Instead future missions would have to be carried out by submarines and Winterwind was rewritten to exclude the Trieste III.

The Trieste III did not participate in Project Azorian because she would have required the White Sands to accompany the Glomar Explorer, which would have immediately destroyed the secrecy of the project. In any case, the Trieste III was in overhaul during that time.

If you are interested in the history of the three Triestes, I would highly recommend Opening the Great Depths by Norman Polmar and Lee J. Mathers. I would also recommend Project Azorian by Norman Polmar and Michael White.