I've heard Christians assert that you can prove that Jesus Christ is the Messiah from the book of psalms. There doesn't seem to be any watertight argument for the existence of messianic psalms other than the New Testament says so, let alone that Jesus Christ is the Messiah referred to in the psalms.
The interpretation of the Old Testament in the New is an extremely complex subject and I recommend the book referenced in this summary article about the perspectives involved. However, there is significant support from Jewish sources for reading at least a some, but not all, of the Psalms as Messianic apart from the New Testament and according to Jewish (non-Christian) sources - which to be fair was largely written by Jewish writers as well.
Exhibit A: Psalm 2
Cited several times in Acts, Hebrews, and Revelation, Psalm 2 has been interpreted as Messianic by the Babylonian Talmud, Maimonides, and the Midrash on Psalms just to name a few. That these sources do so after the first century suggests that in spite of the temptation to interpret them apologetically as being non-messianic, it is a genuinely Messianic Psalm.
Exhibit B: Psalm 110
Although less conclusive than Psalm 2, an article entitle "Is Psalm 110 a Messianic Psalm" says that (footnote 32):
...even the first-century Jewish leaders considered the reference to [my Lord] in Psalm 110:1 to be prophetic of Messiah and to be used only of the Messiah and not of an earthly Davidic king...
This is especially significant because this is the verse most cited by Jesus himself to prove his Messianic identity.
Exhibit C: Psalm 22
On the other side, a previous discussion (here) seems to suggest conclusively that Psalm 22 has not been interpreted Messianic-ally by Jewish interpreters.
These are just three examples that show there is diversity in the way 'Christian' Messianic Psalms were viewed outside of Christianity. But two final points are worth noting:
(1) As "Dan" in the discussion on Psalm 22 notes, it is often difficult to find actual pre-Christian Rabbinic commentary which means that (a) there is apologetic incentive in later sources not disagree with NT interpretations and (b) significant potential for disagreement about what first century Jews really thought. See this article for the dates of Rabbinic literature.
(2) Considering Jesus, Paul, Stephen, and most of the early church were Jewish - some of them highly educated in the Law - so while we can assume bias on their part, this should not cause us throw out their testimony whole-sale, or discount its Jewish-ness. In fact Acts 17:11 tells us it is reading the scriptures eagerly that convinced the Bereans and many others of the validity of Jesus' claims to be the Messiah. This was before the writing of the NT and therefore means it was through reading directly from the Jewish scriptures that they were convinced.
For several books of the NT it can be shown very easily that argumentation for this was introduced retroactively. The clearest example is found in Hebrews, where the author quotes the same prophecy of Jeremiah 31:33 very close to one another twice, but rendered differently:
ὅτι αὕτη ἡ διαθήκη ἣν διαθήσομαι τῷ οἴκῳ Ἰσραὴλ μετὰ τὰς ἡμέρας ἐκείνας, λέγει Κύριος, διδοὺς νόμους μου εἰς τὴν διάνοιαν αὐτῶν, καὶ ἐπὶ καρδίας αὐτῶν ἐπιγράψω αὐτούς, καὶ ἔσομαι αὐτοῖς εἰς Θεόν καὶ αὐτοὶ ἔσονταί μοι εἰς λαόν.
For this is the covenant that I will make with the house of Israel after that time, declares the Lord: I will put my laws in their minds and write them on their hearts. I will be their God, and they will be my people.
Αὕτη ἡ διαθήκη ἣν διαθήσομαι πρὸς αὐτοὺς μετὰ τὰς ἡμέρας ἐκείνας, λέγει Κύριος· διδοὺς νόμους μου ἐπὶ καρδίας αὐτῶν, καὶ ἐπὶ τὴν διάνοιαν αὐτῶν ἐπιγράψω αὐτούς,
This is the covenant I will make with them after those days, says the Lord: I will put My laws on their hearts and write them on their minds
Why this difference? Because in the first time the prophecy is quoted, it is introduced by:
But finding fault with His people, He says: Look, the days are coming, says the Lord, when I will make a new covenant with the house of Israel and with the house of Judah — not like the covenant that I made with their ancestors on the day I took them by their hands to lead them out of the land of Egypt. I disregarded them, says the Lord, because they did not continue in My covenant.
So for the first quote, it is rendered as a criticism against the House of Israel, while in the second quote, it is rendered as a possible promise to all. The second rendering is retrofitting to enable interpretation as fulfilment in the NT.
As for Psalms, if you look, for example, at Psalm 8:4-8
4 What is man, that thou art mindful of him? and the son of man, that thou visitest him?
5 For thou hast made him a little lower than the angels, and hast crowned him with glory and honour.
6 Thou madest him to have dominion over the works of thy hands; thou hast put all things under his feet:
7 All sheep and oxen, yea, and the beasts of the field;
8 The fowl of the air, and the fish of the sea, and whatsoever passeth through the paths of the seas.
And as quoted in Hebrews 2:6-8
6 διεμαρτύρατο δέ πού τις λέγων Τί ἐστιν ἄνθρωπος ὅτι μιμνῄσκῃ αὐτοῦ; ἢ υἱὸς ἀνθρώπου ὅτι ἐπισκέπτῃ αὐτόν;
7 ἠλάττωσας αὐτὸν βραχύ τι παρ’ ἀγγέλους, δόξῃ καὶ τιμῇ ἐστεφάνωσας αὐτόν,
8 πάντα ὑπέταξας ὑποκάτω τῶν ποδῶν αὐτοῦ. ἐν τῷ γὰρ ὑποτάξαι αὐτῷ τὰ πάντα οὐδὲν ἀφῆκεν αὐτῷ ἀνυπότακτον. νῦν δὲ οὔπω ὁρῶμεν αὐτῷ τὰ πάντα ὑποτεταγμένα·
6 But one in a certain place testified, saying, What is man, that thou art mindful of him? or the son of man that thou visitest him?
7 Thou madest him a little lower than the angels; thou crownedst him with glory and honour, and didst set him over the works of thy hands:
8 Thou hast put all things in subjection under his feet. For in that he put all in subjection under him, he left nothing that is not put under him. But now we see not yet all things put under him.
In the Hebrews quote, the "inconvenient" animal references to Genesis are deleted, so that the quote is molded to be interpretable as applying to a messiah by "man" and "son of man".
However, in the original Hebrew text, the words adam and "ben'adam" mean (children of) human being(s) of either gender. In the Greek rendering, the author takes advantage of the dualistic form of the nominative singular masculine ἄνθρωπος and the genitive singular masculine ἀνθρώπου, while there is no reason to do so other than fitting the quote to fulfil messianic prophecy. With those alterations, verse 6 and onwards is turned from a description of what was already done to man into a prediction.