Defender Josh Drummond is also grounded with a similar injury while key position players Jason Roe and Mitchell Clark have recovered from theirs to work towards their first senior matches of the season.
Disturbed by the trend, Matthews has consulted the Lions’ medical staff to source the cause of the problem.
"We don't know the answer," he said on Friday. "I wish we knew the answer. We've searched for the answer. We review everything we do constantly … I ask that question constantly, 'What's going on with our quads?'"
While Begley could spend up to four weeks on the sidelines, having injured himself at training, small forward Ashley McGrath has been omitted from the team that lost to Hawthorn by 12 points last week.
The 24-year-old overcame an ankle injury which hampered his start to 2008.
"He's just not fit enough at the moment and we think he'll get fitter quicker in the reserves unfortunately," Matthews said. "We need his talent but we need him fit and well."
Young guns Scott Harding, Lachlan Henderson, Tom Collier, James Polkinghorne and Bradd Dalziell are all in the selection frame for Sunday’s game, with Collier, Polkinghore and Dalziell in a race to make their debuts.
"If there's one first-gamer that will be tops, we certainly won't be taking three or four first-gamers … you never play a whole lot of first-gamers at one time,” Matthews said.
Matthews didn't back down on his criticism of former Demon Travis Johnstone, who joined the Lions at the end of last year, revealing the 27-year-old had reacted well to his analysis of his performances so far.
"He's fine I think, I basically said I think he could play better," Matthews said.
And the absence of Melbourne captain David Neitz with a serious neck injury will force Matthews to re-assess his defensive structure, though he's pretty sure who will line up on the dangerous Russell Robertson.
"Joel Patfull's done quite well on Russell Robertson a couple of times so that's a good match-up."
