It’s hard to say but most probably not.

Certainly we know folks did love his work as his plays ran for more than two or three performances but in the same measure, Shakespeare never thought about publishing his plays (which was done after his death).

Without the tradition of reading the plays the impact wasn’t upon quoting him as it is today, more upon the immediate impact of watching them.

So quoting him would be less likely than mentioning his stories or the performances.

Included in the Chamberlains/Kings Men were the likes of famed comic Burbage who would improvise lines for comedic effect, while the author himself would take the stage (although he did once comment that some found his accent too strong to understand him at times).

I’ve answered questions on contemporary Shakespeare reactions a few times, here, here and here which I hope place the reactions to his work into context.

So I don’t think their reaction would be the same as ours. Certainly within Romeo and Juliet for example, the fame of the Balcony Scene only really began in the Victorian era some 300 years later.

Hope that helps. Sorry it’s just a brief overview.