Maybe, maybe not.
Case for: Votes Lab has won since 2010, & particularly 2015, tend to be younger, more socially liberal, more concerned about environment & (crucially) have much lower partisan attachment to Labour party. So they're likely to be flightier when unhappy with Lab https://twitter.com/chriscurtis94/status/1358712068954853378

Greens are the obvious place for such voters to go - as the party of "all the things people like me like, and none of the compromises/half measures of Labour". Likely to be particularly appealing to those attracted primarily by Corbyn, whose appeal was also "no half measures".

OTOH, as we saw in 2019, first past the post has a hell of a squeezing impact come election time. Lib Dems peaked at low to mid 20s in summer 2019, ahead of Lab on some polls. then collapsed like a punctured bouncy castle as voters forced to focus on binary first past post choice

In addition, Green voters are likely to be most concentrated in places Lab has least to worry about - ethnically diverse, graduate heavy big city seats where Labour majorities are currently very large indeed on average.

However, election day is a long, long way away, and so the primary political impact of the Greens is likely to come via internal pressure on Starmer and his team. A big Green surge would reverse much of Lab's poll gain since 2019 and make internal critics louder.

This could enable 2021 Lab to Green defectors to exert influence in a similar way to 2011-14 Con to UKIP defectors - protest defections to a smaller party could force a shift in the positioning of their old party by empowering internal critics who think like the smaller party.