Messengers spoken of in the Quran

Many messengers have been sent by God. Only some of them have been mentioned by name in the Quran:

We have sent messengers before you, some of whom We have narrated to you and some We have not narrated to you. 40:78 (also 4:164)

The Quran tells us that God sends a messenger to every nation/community:

For every nation is a messenger. 10:47

Since nations will continue to exist until the end of the world, there is no reason to suggest that there will be no future messengers. It must be made clear that the Quran confirms that Muhammad was the last prophet but not the last messenger (33:40).
For the difference between a prophet and a messenger, please check this page:
Difference between prophet and messenger

The Quran lists 20 prophets by name, who by definition, are also messengers.

Verses 83-86 of Sura 6 lists 18 of these prophets, they are:

1- Abraham (also named as a prophet in 19:41 and 33:7)

2- Isaac (also in 19:49 and 37:112)

3- Jacob (also in 19:49)

4- Noah (also in 33:7)

5- David (also in 17:55)

6- Solomon

7- Job

8- Joseph

9- Moses (also in 19:51 and 33:7)

10- Aaron (also in 19:53)

11- Zachariah

12- John (also in 3:39)

13- Jesus (also in 19:30 and 33:7)

14- Elias

15- Ismail (19:54)

16- Elisha

17- Jonah

18- Lot

Two further prophets are mentioned elsewhere in the Quran:

19- Idris (19:56)

20- Muhammad (the "seal of the prophets" in 33:40)

We have a total of 20 prophets mentioned by name in the Quran.

The Quran also speaks of three messengers (not named as prophets) they are:

Hud, Saleh, and Shu'aib (named as messengers in 26:125/ 143/ 178 respectively)

This brings the total of messengers mentioned by name in the Quran to 23.

In addition to the 23 messengers, we read in the Quran of other men who are praised and spoken highly of, but not named as messengers, they are:

Adam (described as one of the chosen 3:33, but not called a messenger in the Quran)

Zal-Kifl (described as steadfast, patient and righteous 21:85 and 38:48)

Luqman (described as one endowed with wisdom 31:12)