No they don't. I have just gone through the full quote process to double check.
They ask for non-motoring convictions as follows;
Do you have any motoring offences in the last five years, including fixed penalties, convictions, driver awareness course or disqualifications or have any pending prosecution?
Furthermore, their T&C's state there following:
We may also collect information defined as "sensitive data" within the Data Protection Act 1998. This includes:
Criminal convictions etc.
However, this is not asked for during the quote process, nor when taking up the insurance. Remember they must specifically ask for the details as the onus is on them to establish all material facts.
Of course, if you know of information contrary to this, please let us know where it is found.