The gospel of Luke and Acts are believed written 60-62 AD. The Letter(gospel) having been written first.
Paul wrote Timothy and Titus after his realease from prison 62-64AD
The letter written by Luke would have had wide circulation.
There are some that believe that Pauls use of the word "scripture" in that passage is meant for both the OT quote and Luke...But that is not as important a point as the fact he quoted it with authority
SD