Does a turret going to "hostile" state, but not opening fire, disqualify the user from Foxiest of the Hounds?

No, as long as an alarm didn't go off, you should be safe.

Quotes from the Deus Ex wiki:

This does NOT mean that you can never get an "Alerted" status from enemies, or even "Hostile" with open combat. It simply means that no alarms can be tripped, such as from a camera spotting you or a body, an alerted enemy touching an un-hacked alarm panel, etc.

It is wise to upgrade your capture software to enable you to shutdown turrets and bots; while getting their attention does not cause an alarm directly, the gunfire will get anyone's attention in the room.

Source

Areas with alarms have those red lights dotted all over the map. When those start to flash, it means the alarm has been triggered. This is how I usually tell, as sometimes I might accidentally set off the alarm without even knowing.

Just to answer your question, under normal circumstances, as mordi2k has mentioned, as long as a turret doesn't start shooting at you, then you are safe. However, in this case there is actually no Smooth Operator bonus for the first mission in the Sarif Manufacturing Plant, hence no danger of breaking the Foxiest of all Hounds achievement to start with.

You might want to read the answer I posted on this thread:

Which missions can I earn the Ghost and/or Smooth Operator bonus on?