If I have a Razer Blade 15" laptop with a 60Hz display. If I connect an external monitor with a 120Hz refresh rate via HDMI, can I use the 120Hz refresh rate while gaming on the external monitor or will it run at 60Hz?
You haven't mentioned some other specifications of the laptop or monitor that might help to give you an answer, but from what I can tell, the laptop has an HDMI 2.0 port which is capable of driving a 1440p display at 144Hz (it could even go up to 240Hz, but at a worse Chroma Subsampling). Of course, if it can run a 1440p monitor, it can definitely run an HD one at those frequencies. That laptop also has a Mini DisplayPort 1.4. That is more than capable of driving an HD, 1440p, and even a 4k display at 120Hz (just barely at 4k however).
I'm going to guess you have the GTX 1060 model of the laptop. If that is the case, using an external monitor within HDMI 2.0's capabilities should work with no issues. Bare in mind that higher resolutions with higher refresh rates are going to demand more from the hardware however, and could cause some poor performance in workloads if you are pushing the hardware too hard.
You can check/change what refresh rate your display is running at using Nvidia Control Panel - something you should have with that laptop since it has an Nvidia GPU. Simply go to "Adjust desktop size and position" and take a look at the Refresh Rate control.