What is the best way to connect four 4-line analog phones to trixbox ?

Any solution you can use from an ata to using a 4 line fxs card will cost way more than a VOIP phone. The best solution in this situation is to install a voip phone next to their exist analog phone. That way they can transition slowly.

Some clients are worth keeping, some worth forgetting.
Although worth forgetting, but customer still is king.... although many of them orders and demands, well, indeed act like ---- Kings!

Merry Christmas to all!

This one smells like never ending trouble. My people had the same phones but they didn't ask to keep them thankfully. If your users want to use them I'll bet they want to use the proprietary intercom feature which depends on having line 1 at minimum in parallel. They'll never use more than a fraction of the PBX features and won't notice any benefits (but complain a lot). However to use them like they are you only need one 4 port FXS and then parallel each port *4 making each phone identical except for their own proprietary ID. The default line option on the phone would pick the appropriate extension. I can think of a few more things they will complain about. Rather a waste of a good pbx.

You could more easily simulate what they have now by using even a cheapish 4 line digital phone with BLF buttons. They will have to learn just enough to notice the added functionality and be happier in the long run.

What's worse than an unreasonable client is be just an employee that is responsible for doing the same job internally. They have cheaper budgets and higher expectations. And you will hear about it forever and always while doing regular work.

What else do they expect to accomplish besides VoIP? Are they happy with functionality as it is and only want to lower costs? What else do they want it to do?
The cheapest thing to do without changing any perceptions (until there are internet or net issues) is to ATA/FXS some VoIP lines, keep one landline on #1, and hook it up to what is already there. Simply replace 3 landlines with VoIP lines and change nothing else. no PBX, just pick the right voip provider. Use a stand alone FXS device with failover to the landline on #1. You'll make a better margin in a shorter time and they start saving usage costs without any change in functionality until they decide to use some modern features.
my $10% advice...

If you have a revenue stream and can afford it there is nothing wrong with leasing a system. Since we are also the Voip provider we supply the server as part of the contract, the customer can either buy or lease the endpoints.

Support and setup costs are the same, you simply factor them into the lease.

Here are the current lease factors we use to calculate monthly payments. If the customers credit is shaky you can always bump it up a bit. These provide about a 16% rate of return on your money assuming a 30% value at end of lease. These are not $1.00 buy out rates. Remember you accrue all the depreciation of the leased assets during the lease term.

24 Months: .04758

36 Months: .03375

48 Months: .02690

60 Months: .02284