Signing his first professional contract with Queens Park in 2016, Liam made his debut in a 3-0 home defeat to East Stirling in February that year. After finishing 4th place in the overall standings that season, Queens faced Cowdenbeath from the division above in the playoffs. A double from Chris Duggan in the first leg was enough to take Queens Park through to the final after a 1-0 defeat at Central Park.
Two games away from promotion, only Clyde stood in their way and they got off to a great start after a 3-1 victory in the first leg. Clyde fought back in the second leg and even though were eventual winners on the day, Queens were promoted to League One after a 3-2 victory on aggregate.
The following season Brown made 23 appearances for Gus MacPherson's side with 16 of those in League One. After impressive performances on the pitch, Motherwell announced the signing of Liam in June 2017 initially to play in their under 20's side. He earned his first-team debut as a substitute against Hamilton on 12th May 2018, where they ran out 3-0 winners.
Since then, Liam has joined Scottish League Two side Edinburgh City under current boss James McDonaugh and was a regular starter in the side up until the season was postponed in March due to the coronavirus pandemic. Edinburgh eventually finished runners up to Cove Rangers and hope to go one further to earn promotion next year.