Spend a penny is an English euphemism for using a lavatory. The phrase originated when public toilets in the UK used to use coin-operated locks and you would spend a penny in order to use the lavatory. Nowadays, the phrase is used as a euphemism so to be polite when saying someone is using the bathroom.
In Ted Lasso, Keeley said to Higgins and Rebecca Welton that she was going to spend a penny ready for the flight to Everton F.C. home stadium, Goodison Park.