The answer to that will partly be decided by what your bees do in the various seasons. Your climate looks pretty similar to mine, and if that is the case, I bet your queens stop laying drones in around March (mine stop in early September). This means that a couple of weeks after mid-summer, virgin queens may find it difficult to get well-mated, as there won’t be a lot of drones around.
According to my mentor, his bees start making drones again in mid to late December (mid to late June for you) and swarm season starts about 6 weeks after that. Swarm season won’t start until there are plenty of drones for mating.
Putting this all together, and allowing time for drones to mature and accumulate numbers, I would say that trying to raise a new queen between February and July in your climate, may result in a poorly mated queen. I may be totally off the mark, but it seems pretty likely based on your climate.
Having said all that, I can get beautiful queens from Hawaii year-round. The climate there is tropical, of course, so if you have access to somebody who can rescue you with a queen from a sub-tropical or tropical apiary, you have options.
Your friend is totally correct, by the way. First year queens rarely swarm if you give them enough space. Second year queens swarm more frequently, and third year queens even more so. Doesn’t mean yours will, but it becomes more likely as time goes by, unless they supersede her first.