Beere raced ahead and scored the first 12 shots without reply.
Solway managed to convert the odd single, but Beere’s fine drawing shots carried her to a 21-6 victory and she will now face Gary Pitschou in the semi-final in August.
Coming off the back of his Rothschild Open Singles Tournament win and keen to finally get his name on the Open Singles trophy, Pitschou went through against
fellow Vale Rec bowler Nigel Collins.
Nothing separated the pair at the 10-end mark, but from 8-8 onwards it was Pitschou who controlled the match and with a string of doubles and triples won 21-12.
Matt Le Ber and Terry Brokenshire make up the semi-final line-up.
Le Ber played the only other female quarter-finalist, Lyn Small, and, playing well, triumphed 21-8.
He now faces Guernsey Bowling Club’s Brokenshire who was runner-up to Pitschou in the Rothschild tournament.
Brokenshire dominated Steve Williams from the Vale Rec to lead 18-4 after just 11 ends, before Williams started to dig in and recover.
But invariably he was reduced to singles and Brokenshire went through 21-12.
In the men’s under-25 singles, Vale Rec’s Kris Lowe earned himself a semi-final clash with Ben Harvey after defeating NBA’s Kieran Queripel 24-13 at the Delancey green.
It was close until Lowe scored two full houses back-to-back.
In the Smithard Over 60s singles, defending champion Len Le Ber continued his undefeated record in the event.
Having turned 60 at the beginning of last year, he has not lost a game so far, including the Channel Islands final played at the start of the season.
This time he defeated Vale Rec clubmate John Rihoy.
It was a close match to start with being 8-8 after seven ends, but Le Ber then recorded a a full house of four, which proved to be the winning margin.
Don Ogier made easy work over Keith De la Mare returning a 21-6 card and will now play Le Ber in one semi-final, while in the other Williams beat Stan Simon 21-14 to clinch a semi against Brokenshire.
The GBC man beat Francis Tardif 21-14.