Sure. What you have is a device that takes multiple HDMI inputs (4) and sends any of those inputs to 2 outputs. It's called a matrix switch. The SolidMate brand was sold by Monoprice.
To use the switch basically you need to tell each output which input to use. There are two buttons to select the output, but I don't see any input buttons. So, I suspect (but don't know) that you just keep pressing the output button (A or B) until you select the input you want to see.
The only thing I can't figure out is that you said there were 4 TVs. The matrix switch you are showing only has two outputs, so there must be some other type of device for the other two TVs.
One way to tell is to turn on all 4 TVs and see if three of them change at the same time when you select something different on the A or B output. That would mean 3 TVs are tied to one output using a distribution amp (also known as a splitter). Of course it could also be 2 TVs on each output with two distribution amps (splitters).