Little is known about Velma other than she's elderly and lives in Arlen. She first appears in the Season 2 episode "Snow Job". In this episode, she calls Hank to tell him she's become frozen to her walker after running out of propane to heat her house. She's seen at the end of the episode with icicles hanging from her walker as she cheers on the propane trucks.
Velma makes another appearance in the Season 2 episode "Junkie Business". Buck brings her to Strickland Propane to tell her about Strickland's new electronic payment system. She then encounters Leon Petard as he's foaming out of his mouth due to drug use.
Lastly, Velma is mentioned in the Season 3 episode "A Fire Fighting We Will Go" as Bill tells the guys that she's looking for four plumbers. Included in the compensation for the work is all the beer you can drink, which convinces Hank to accept the job.
She is not seen or mentioned after this episode for the remainder of the series.