Custom links and custom buttons are stored as a WebLink, which is a subcomponent of Custom object. If you wish to compare or deploy them, you need to ensure Custom object is included in the Gearset metadata comparison filter.

You can choose to compare the entire custom object with all of its subcomponents, or use the metadata filter to narrow down the retrieval to only include WebLink.

On the comparison results page you can quickly narrow down your results to only show your new and/or changed WebLink objects by filtering the Metadata type column as shown below.