So would I have to open the computer? Would I be able to tell as I have never built a computer?
I wouldn't recommend checking thermal goop to someone who was unfamiliar with the insides of a computer. Even for a computer that was not under warranty.
Thermal goop goes under the heat sink, which has a fan on top.
There's no way of checking it without removing the heat sink, and then you'd need to reapply the goop. That can be tricky. You have to avoid bubbles, and use the right amount of goop. Too little and you get insufficient coverage. Too much and the goop oozes out from under the heat sink and gets on other stuff where it shouldn't be.
Many companies don't even use thermal goop unless you specifically request it.
Instead they use thermal pads that are easier to apply.
If you suspect a problem with your computer and it's under warranty, you probably want to contact their tech support. Homer6 may be right about the problem being related to processor cooling, especially if the computer does this while you're using it and it's not caused by sleep or hibernation settings. But I wouldn't recommend a newbie try to fix something like that themselves without supervision by someone familiar with applying thermal compound.
As Draclvr says, you can tell whether the processor is cooling properly by monitoring processor temperature.