I salvaged a 4v Li-po battery out of a broken Bluetooth speaker, it has what appears to be a charging circuit under the battery wrap. Will I be able to charge this battery from a computer 5v USB connection without any problems?

 

How was the speaker charged?

 

If it's the circuit under the kepton yellow wrap ...no....that is a safety ic most lipo's come with.
You usually need a charger circuit.
Aliexpress has the quite cheap.
Never charge a lipo with a current exceeding it's mAh capacity.

 

That is almost certainly only cell protection, but, because it includes over and under charge protection it is often the only battery management in devices that use a single prismatic LiPo cell.

A 47Ω 1/2W resistor in series will limit the current to ≲125mA with 5V applied. The protection board will stop the charging then the cell reaches ≲4.2V.

To be sure the board is operational, I would test this with a bench supply or sacrificial USB wall wart before trusting it, on say, a computer USB port.