as mentioned by Jacopo, that depends on the region and position you are applying for. I will refer to Europe/US in the following; the calendar should be moved accordingly in other geographies to take into account the equivalent of Summer/Winter breaks.
For new graduates, usual starting dates in Europe/US are September and January. Statistically, many interviews take place in September-November (January intake) and January-March (September intake). The best period to apply as a new graduate is the first month of the period when interviews take place, since the process just reopened and you will have less competition. Thus, that would be September for the January intake, and January for the September intake. Having said that, you should check for the specific deadlines of the programme you are interested in, as they may vary by country. The best period may also depend on on-campus recruiting, which for undergrads is usually in September-October and MBA in January-February.
If you are not applying as a new graduate, there may be a different timeline. For summer internships you can apply as early as September/October for the year after (here as well, the earlier, the better for competition). For experienced hire positions you can apply through the whole year, as the demand for that will depend on specific projects (ideally, still avoiding July and August for the summer break). In some countries, bonuses may have an influence on the period when to apply as an experienced hire, as more people than usual may leave after receiving them (usually in March); thus the firm may have a lack of experienced consultants right after that and increase intakes.
In terms of intakes: the largest is usually the one in September, however you usually have more competition for that as well. Intakes may also be unpredictable, as they may depend on how many projects are starting. Thus, if you are ready to apply in September/October to start in January, it doesn’t make much sense to wait the September intake.
In terms of projects: as mentioned by Jacopo the busiest months are September and January; quality would be more difficult to define, as it would depend on your specific preferences.