We all know that the Quran is our major source for anything about islam, but it is more likely to be a kind of abstract this means when it comes to details we need the sunna as a 2nd reference.
Now let's see what we can find in Quran:
Maintain with care the [obligatory] prayers and [in particular] the middle prayer and stand before Allah , devoutly obedient.
[Surat al-Baqara (2:238)]
And establish prayer at the two ends of the day and at the approach of the night. Indeed, good deeds do away with misdeeds. That is a reminder for those who remember.
[Surat Hud (11:114)]
O you who have believed, let those whom your right hands possess and those who have not [yet] reached puberty among you ask permission of you [before entering] at three times: before the dawn prayer and when you put aside your clothing [for rest] at noon and after the night prayer. [These are] three times of privacy for you. There is no blame upon you nor upon them beyond these [periods], for they continually circulate among you - some of you, among others. Thus does Allah make clear to you the verses; and Allah is Knowing and Wise.
[Surat an-Nur (24:58)]
Verse 1 tells us that the number of prayers will be odd and this means we have at least 3 prayers.
Verse 2 tells as that during daylight we have 2 prayers (duhr and a'sr) and one when the night approaches (al-maghrib) (see also this Fatwa in Arabic): With (only) this information it's still possible that we only have 3 prayers, but
Verse 3 which teaches us about the resting times mention clearly the fajr/sobh prayer and the I'sha prayer as the fajr itself is a not mentioned time in the 2 Verses before we find out we really have 5 prayers mentioned in the quran!
Now we find we have 5 prayerers or prayer times mentioned in the Quran.
By the way we could also include some other Verses to explain for example:
Establish prayer at the decline of the sun [from its meridian] until the darkness of the night and [also] the Qur'an of dawn. Indeed, the recitation of dawn is ever witnessed.
[Surat al-Isra' (17:78)]
Which mentioned indirectly the prayers from duhr until i'sha' and by the mentioned recitation of dawn the fajr prayer! See also thisFatwa in Arabic
Just as an addition
the sentence أقِمِ الصلاة = establish prayer (singular form) was quoted 5 times in the Quran. The sentence أقيموا الصلاة= establish prayer (plural form) was quoted 12 times in the Quran.
As we know 12+5=17 which is the sum of the raka'at of our daily fard prayers!
The Verse 1 from above is written in the Moshaf as follows حَفِظُوا عَلَى الصَّلَوَتِ وَالصَّلَوةِ الْوُسْطَى وَقُومُوا لِلَّهِ قَنِتِينَ
if we count the number of Arabic letters at the right and left from the expression وَالصَّلَوةِ الْوُسْطَى = and the middle prayer
We find it to be the exact middle of the Verse and each part at the left and right has 14 Arabic letters and if we ignore the independence of the letter "وَ" before "قُومُوا لِلَّهِ قَنِتِينَ" (and this makes this count a bit weak in my Opinion) we could say that each part left right and middle of this Verse consists of 3 words and again 3+14=17.
And Allah knows best