Does Paul Quote Luke?

Some Christians argue that the Gospel of Luke was written early because (among other things) it is cited by Paul.

The labourer is worthy of his reward

1 Tim. 5:17 Let the elders that rule well be counted worthy of double honour, especially they who labour in the word and doctrine.
18 For the scripture saith, thou shalt not muzzle the ox that treadeth out the corn. And, The labourer is worthy of his reward.

Here it is in Luke:

Luke 10:7 And in the same house remain, eating and drinking such things as they give: for the labourer is worthy of his hire. Go not from house to house.

This argument has a number of issues...

The Epistle was not written by Paul

Most modern scholars believe 1 Timothy to be pseudographa, written later first century or even second century. This gives it plenty of time to quote Luke.

A Compilation

The author of Luke makes no claim to be an eye witness, but instead is a compiler. He has collected various texts and accounts and woven them into one book. It is entirely possible that Paul was familiar with any number of those older works, and that he and the author of Luke were both quoting the same earlier document.

Scripture

Or it would be possible, except that the verses say the quote comes from scripture. For Paul, that meant the Old Testament, and only the Old Testament. If it was Paul who wrote the letter, then he is not citing Luke, but the Hebrew Bible, perhaps one of these:

Leviticus 19:13 Thou shalt not defraud thy neighbour, neither rob him: the wages of him that is hired shall not abide with thee all night until the morning.

Deuteronomy 24:15 At his day thou shalt give him his hire, neither shall the sun go down upon it; for he is poor, and setteth his heart upon it: lest he cry against thee unto the Lord, and it be sin unto thee.

This is my body, which is broken for you: this do in remembrance of me

This one is a little different. Here it is from Paul:

1 Corinthians 11:23 For I have received of the Lord that which also I delivered unto you, that the Lord Jesus the same night in which he was betrayed took bread:
24 And when he had given thanks, he brake it, and said, Take, eat: this is my body, which is broken for you: this do in remembrance of me.
25 After the same manner also he took the cup, when he had supped, saying, this cup is the new testament in my blood: this do ye, as oft as ye drink it, in remembrance of me.

And here in Luke:

Luke 22:19 And he took bread, and gave thanks, and brake it, and gave unto them, saying, This is my body which is given for you: this do in remembrance of me.
20 Likewise also the cup after supper, saying, This cup is the new testament in my blood, which is shed for you.

1 Corinthians almost certainly was written by Paul, and he is not saying this is from scripture, so perhaps this really is Paul quoting Luke?

It is possible, but these quotes are equally evidence that Luke was quoting Paul, or both were quoting an earlier source. This is a long way from proving Paul copied it from Luke.

Remember that Paul did talk at some length to the disciples, and if the last supper really happened, it is just not credible to suppose the disciples never told Paul about. The argument that Luke was first really only works if we assume Luke made up the story. I doubt any Christian wants to go there!