Actually, currently there is no way to upload your own icons, just the system ones.
However, we already have a feature request in order to allow this in the feature which is tracked on the issue below:
I would advise you to add your comments to this feature, and also vote for this implementation.
For more information about how Atlassian implements new features, please, check the document below:
...word of mouth, so by 2016, we were working with several other entities on campus to implement Jira Service Desk . The Atlassian motto of “for every team” has really come true for us in this case. We...
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