It’s a versatile usage that often collocates with get and is typically associated with moody or petulant grumpiness, and sometimes with drunken antagonism. I heard the phrase regularly when I was growing up in the rural west of the country, and now in the urban west I still do, occasionally, in expressions such as:
She’s fierce thick over it. [fierce = very]
Don’t you be getting thick with me.
He got pure thick about it.
Note that “Thick out!” just means “Thick!”: “The out serves as a mild intensifier and colloquial marker.” Stan has many lively quotes, and it’s interesting that he used Twitter to collect them. I envy the Irish for growing up with such a trove of expressive English.