On the one hand, waiting till your alma matter acquaintance suggests the idea, might lead to it never happening. On the other, being pushy and putting him or her in an awkward situation gets you no further.
Having previously been in the awkward position of having a few multiple former classmates, not all of whom were all-star performers, persistently ask me to forward their resume to my boss, I would hesitate. You might want to ask your contact whether s/he thinks you'd be a good fit for her company. If she has any reservations, ask lots of questions, and give him/her some ammo to sell him/her on the fit being good.
Do all of that before suggesting s/he approach his/her boss, so you get a glowing recommendation rather than "apologies in advance boss, another classmate asked me to forward you their resume, I don't even know this guy!". Even if she tries to be polite, if she's curt and enthused in her language on one forwarded resume, and glowing in another, the boss may realize this was sent "just as a favor" and not follow up. Your acquaintance certainly doesn't want to risk tarnishing her reputation by endorsing an unknown quantity that turns out to be a bad hire, so make sure s/he has gotten to know enough about you to feel confident recommending you.
You could broach the topic by asking whether s/he gets people asking to forward their resumes all the time? Is that super-awkward? "well, then I probably shouldn't ask, should I?" ;-). Ask her/him whether it's better to apply through internal channels (like forwarding her resume) or through the formal channels (it depends on the company!). Give her an easy out if she's really uncomfortable with the idea of forwarding your resume, but at some point, if you don't ask you will not receive.
Consider, also, other alternatives. Would it be just as good to take her and her boss out to lunch sometime so they can get to know you better? Introduce you to someone in the department you're looking to apply to? Etc.