Basically it's a process that estimates the number of hours effort...
Most clients will come to you with a set of requirements - these may be very clearly thought through or it might be a vague idea that is still in somebody's head.
The first meeting you would normally do for free, so they can gauge your style/communication/project management methods. They might even ask you to pitch for the job against other design agencies. You need to know what you are doing if you're going to pitch against other agencies. Lets not go there for now.
Assuming the client has a bunch of requirements, after the first meeting where you understand what they are trying to achieve you would probably do a little bit of work to demonstrate to them that you are the man for the job. This may not even involve design - it might just be clarification of their goals written down in a formal specification.
Most leads will involve some risk of the potential client not becoming a client - so you have to be prepared to lose some. The more engaged you can be with them and their idea, the better the chance that you will get their business. Show them other relevant work you have done to convince them you're the best choice. It's likely they are looking at other design agencies as well. Find out what their budget is.
Once they decide they like you (but before you quote), it's back to the requirements and breaking it down into milestones or deliverables. This is basic project management really. Talk to them about the milestones and break each one down into a list of tasks. At the task level, estimate how long each task will take. At the end of this process you will have a bunch of milestones and tasks and an understanding of how many hours it will take for this to happen. If you are not familiar with estimation (or even breaking the milestones down into tasks) you may have to outsource this (or employ a project manager). Try to understand from the client which milestones (and what functionality) are must have and which are nice to have.
Throw some contingency into the estimation - between 10 and 20%, and multiply by your hourly rate. Remember what their budget is. Then go back to the client and see how they react. Most will try to negotiate. You know what your margins are and you know how hooked in your clients are, so make the right judgement such that you keep the business but without reducing your margins too painfully. That's the quote end of things.
If you haven't been involved in quoting for web design work before I'm not sure how you are going to do it other than handing off the whole process to somebody that can do it. Unless it is very simple, there is going to be a lot of to-ing and fro-ing which if outsourced (and you being the main point of contact) will be a laborious and potentially frustrating process for all concerned.