If any SEO firm tells you a specific time frame or give you a guarantee within 1 month, 3 months, etc. than they just have a good sales team. The truth is that there is no definitive answer. That sort of honesty is why the main source of our growth is referrals.
Your results depend on how competitive your industry is. If you sell cars nation wide than that is a lot less competitive than selling a very specific blue widget that is only available in Utah. Your website may rank quicker or take longer than a competitor based on what keywords you want to target, their level of competition, etc. The more competitive your industry is and the broader the audience or location that you sell to than the more in-depth your online marketing campaign needs to be.
Some SEO clients have experienced page one results in as little as three weeks.
However other more competitive, higher valued keywords can take months, or even years. Having a targeted, goal oriented optimization campaign in place with a qualified SEO agency driving the efforts will provide you with the best results.