The standard hotel definition of a stay (at
all hotel programs) is:
One or more consecutive nights at the same hotel, no matter how many times you check in or check out.
It may be worded a bit differently at each hotel program (some confusing leave out "one or more") but they all really work the same way.
Ie, from the time you arrive at the hotel to the time you leave, no matter how many separate reservations it may have been, that's one stay.
Obviously, the fastest way to status with hotels that have a "stay" option is true one-night stays. If you need consecutive nights in the same city, you have to "hotel hop" (change hotels every night) to get maximize the number of "stays" in a period of time.
Having said that, LQ's Gold status isn't worth that much. It's a 20% bonus on points earned on hotel stays, but of course if you choose to earn airline miles (and LQ has some of best airline mile bonuses out there, also on a per-"stay" basis so also best if you're doing one-night stays), that's irrelevant. Its other benefit is two upgrade certificates per year, but the thing is, many LQ hotels don't have any upgrade rooms (or their upgrade rooms are so few that they're always booked up)!
I got up to Elite member (one level above Gold), and still the only benefit I notice (since I choose to earn miles) is a welcome gift (bottle of water and a snack).
The status level benefits are described here:
And if you go their T&C page:
and search for the word "stay", you quickly come up with their definition, which is:
A stay is defined as one or more consecutive nights at the same hotel, even if intermittent check-in/check-out events occur.