How to convince a customer of an offer or service when it doesn't match his inquiry?

If there is such condition then you can honestly tell the customer that what service he/she is searching is impossible to provide at that time but convince him/her to get the specific place to go.
but the conversation should be in the polite way from which you have to give your impression.

Be honest, tell the customer that you do not have the product that he is looking for and provide him with the alternatives and show him the positives for the product that you do have, if you can sell your product, you will have gained a happy customer who will be happy to deal with you again.

Dont even try doing that.
Customer service is to service customer needs and not to change his needs.
Proposing an alternative or substitute that actually does not match his requirement is trying to make a customer spend more reducing his disposable income.
If service does not match his requirement take initiative and find out where he can get what he wants.
It will not only leave the customer with a great experience but also a feel good factor interacting with your brand/company

Sometimes it is nice to lead a conversation with your customer and then listen, it is when you listen that he gives you clues of his needs.
This is when you identify the opportunity, relate and promote your offer linking with benefits to him.
Do not promote the offer, promote the need for your customer to avail the offer.
Am sure any one will buy anything if he "feels the need to".

People, what is happening? Why you force customer ! The issues from the company which is targeting wrong people.
Or has product not fit on customer needs.

It is recommanded to try to offer a service wich match the customer inquiry.
It is better to offer more than one solution.
For exemple, for someone who have a problem with a laptop you can offfer, the replacement of the defect part, a buy-back or a eparation of the defective parts.

understand exactly his needs and work on finding the most suitable solution for him

treat him with respect,serve him with passion and honesty.

ask,ask,ask

always ask to know his needs

It is not ethical to convience a customer with a product that does not match his enquiry but either withdraw from the deal in an eligant way that does not demean your product and start conversation with your company management on how to improve your service according to customer needs if the situation implies such case 

You must have all the information about your product. Tell the customer all its good points (but with confidence), and any free services available. Tell him or her if there is any guarantee of the product. Whatever you say, it should be said in an effective and attractive way. Make sure that you communicate well with the customer. Be polite and patient.

If the customer did not met or match what he is looking the sales representative offer similar to item.

I think in this case i would put myself in place of my customer and ask myself if I am not interested in ths product why should i buy it in the first place .this will help me find out the worries and blocks that would prevent me selling the product and then if I manage to convince myself to buy the product then the customer will buy it too .