There is a difference between a customer who has not paid and a customer who is refusing to. The first is a cash flow problem. The second is a dispute, and it needs handling differently from the moment you recognise which one you have.
Establish what the objection actually is
Ask directly, and ask for it in writing. "Is there something about the work or the invoice you are unhappy with?" It is a simple question and it produces one of three answers.
A genuine complaint about the work. Something was not delivered, or not as agreed. This is a service problem before it is a payment problem, and fixing it is usually cheaper than arguing about it.
A disagreement about what was agreed. Scope, price, or what "finished" meant. This is where the absence of a written quote hurts, and where a written one usually ends the argument immediately.
No answer at all, or a shifting one. That is not a dispute. That is a customer who cannot or will not pay and is looking for a reason, and it should be treated as non-payment rather than as a disagreement.
Separate what is disputed from what is not
If a customer is unhappy with one item on an invoice, the rest is not in dispute. Ask for payment of the undisputed portion while the remainder is discussed.
This is standard practice, it is entirely reasonable, and it prevents a small disagreement being used to hold a large invoice. A customer who refuses to pay any of it, over an objection to part of it, has told you which of the three situations you are in.
Put your position in writing, once, plainly
What was agreed, what was delivered, what is owed, and what you propose. Attach the quote, the acceptance, and the invoice. No adjectives.
Do this even if you have been discussing it by telephone for weeks. A calm written summary is the document everything afterwards is built on, and it very often ends the matter — because it is the first point at which somebody on their side has to write down a reason.
Decide what it is worth
Before escalating, work out the number honestly. What is owed, less what pursuing it costs in fees and in your own time, multiplied by the realistic chance of actually collecting from this particular customer.
Sometimes that arithmetic says pursue it. Often, on a small invoice, it says send a final written demand, decline further work, and put the time into customers who pay. Neither answer is a failure; choosing without doing the arithmetic is.
Keep it civil
Not out of politeness. Because anything you write may be read later by somebody deciding who was reasonable, and because the customer who calms down and pays is not available to the supplier who sent the angry message.
General guidance, not legal advice. Where the amount is significant or a contract is in dispute, take proper advice early.