In captivity, lovebirds could lay eggs at any time of year, if conditions are favorable for breeding. From what I've read, many breeders favor breeding during cooler months and will only allow one clutch annually so the birds have time to recover from the stress of egg-laying and raising young. If given free reign, the birds might breed too often and suffer ill health or even die from complications, like egg-binding or malnutrition. In the wild, seasonal weather changes and other environmental signals interrupt breeding behavior by triggering hormonal changes, so over-breeding is rare. Many of these triggers are absent in captivity, so it is important to practice good husbandry and take steps to discourage breeding when it is not appropriate.
Regarding age, a female lovebird could potentially start laying eggs as young as 9 months, but early egg-laying can be hazardous to a young bird's health, so it should not be encouraged. It is advisable to wait until a lovebird is over a year old before attempting to breed. A female can lay eggs in the absence of a male partner. If there is no male bird, the eggs will be infertile and cannot hatch. If you don't plan on breeding your birds, it is best to avoid stimulating hormonal behavior, like nesting and laying eggs, by not providing suitable nesting locations (no nest boxes or other cavities) and providing ample time for sleep (long nights, like during winter) and changing the diet, if necessary. Under these conditions, the bird is less likely to lay eggs and also will be less likely to display territorial aggression, excessive chewing, cavity-seeking and other undesirable hormonal behaviors.
If you are thinking about breeding, be sure to do lots of research before you start. It is a complex and time-consuming hobby. You will want to arm yourself with lots of knowledge so you are prepared for common problems and can avoid costly mistakes.
As for your last question, I have never tried a lovebird egg, but I assume they don't taste that different from a quail egg.
