Yes, it does.
The rule says that Sleep Clause applies to all kind of sleep inflicted by the target, i.e. non-self inflicted.
A replay to prove that.
Turn 2, when Latios uses Psycho Shift:
Latios is fast asleep.
Latios used Sleep Talk!
Latios used Psycho Shift!
The opposing Heatran fell asleep!
Latios woke up!
Next, on turn 5 when Breloom tries to use Spore on Gyarados when one of its teammate aka Heatran is already sleeping:
Turn 4
Breloom used Spore!
Sleep Clause Mod activated.
So, therefore Sleep inflicted to the target through Psycho Shift counts towards Sleep Clause.
Shout-outs to @SSuperiority for helping me in this test!
EDIT: So you wanted to do in that scenario. I did it again :/ Thanks to @SleepyDreepy for testing second time.
Replay 2
Basically same, this case I did Spore first then Psycho Shift. The result was same again.
The opposing Breloom used Spore!
Heatran fell asleep!
The opposing Latios used Sleep Talk!
The opposing Latios used Psycho Shift!
Sleep Clause Mod activated.
Hope this helps!