I've never seen any evidence for that, and I have studied the man pretty extensively. He's one of my childhood heroes - although I do not endorse his personal sexual choices (and was unaware of them when I was a child and learning about him), he was critical in keeping the world free.
However, the relationship for which he was prosecuted for indecency was with a 19-year-old - a minor in 1952 - and it is quite likely Turing would have not been prosecuted if his partner had been 21 - by the 1950s, it really was rare for the CPS to proceed in 'consenting adults' cases - he clearly had no respect for the law.
The age of (hetrosexual) consent in the UK has been 16 since 1885, having been as low as 12 10 years earlier.