Is this statement supported by Qur'an?
First of all scholars understand the words
And your Lord says, "Call upon Me; I will respond to you." ...
as an order from Allah, which means if you don't do dua' (at all) you are not following this order (Fatwa in Arabic).
Allah also says -as The Z pointed out in a comment-:
Say, "What would my Lord care for you if not for your supplication?" ...(25:77)
The meaning of this verse is also said to be explained in the sahih qudsi hadith:
"O My servants, I have forbidden oppression for Myself and have made it forbidden amongst you, so do not oppress one another. O My servants, all of you are astray except for those I have guided, so seek guidance of Me and I shall guide you, O My servants, all of you are hungry except for those I have fed, so seek food of Me and I shall feed you. O My servants, all of you are naked except for those I have clothed, so seek clothing of Me and I shall clothe you. O My servants, you sin by night and by day, and I forgive all sins, so seek forgiveness of Me and I shall forgive you. O My servants, you will not attain harming Me so as to harm Me, and will not attain benefitting Me so as to benefit Me. O My servants, were the first of you and the last of you, the human of you and the jinn of you to be as pious as the most pious heart of any one man of you, that would not increase My dominion in anything. O My servants, were the first of you and the last of you, the human of you and the jinn of you to be as wicked as the most wicked heart of any one man of you, that would not decrease My dominion in anything. O My servants, were the first of you and the last of you, the human of you and the jinn of you to rise up in one place and make a request of Me, and were I to give everyone what he requested, that would not decrease what I have, any more that a needle decreases the sea if put into it. O My servants, it is but your deeds that I record for you and then recompense you for. So let him who finds good, praise Allah, and let him who finds other than that blame no one but himself."
See also in this article,about this last verse of surat al-Furqan on islamweb in Arabic.
Authenticity of the Hadith
The Hadith you mentioned appears with slightly differences in many Hadith collections, like Sunan ibn Majah (qualified as da'if), Jami' at-Tirmdihi (qualified as da'if) and according to ibn Hajar al-'Asqalani in his comment of Sahih al-Bukhari on this Hadith says it's also narrated by Imam Ahmad, al-Hakim, al-Bazzar and al-Bukhari himself in his book al-adab al-mufrad. So normally as al-Hakim narrated it the narrator chain would be the same as in one of the both sahihs. But as on the other hand al-Bukhari didn't fix it in his sahih, but in his al-adab al-mufrad this could be an issue. One major point is that all these narration chains include Abu Salah al-Khuzy أبو صالح الخوزي, this is a man whom some scholars considered as da'if (like ibn Ma'yn) while others considered him as trustworthy to some extent.
Ibn Hajar added a commentary of at-Tybi in his commentary (My translation take it carefully):
قال الطيبي : معنى الحديث أن من لم يسأل الله يبغضه والمبغوض مغضوب عليه
والله يحب أن يسأل انتهى
"The Meaning of this Hadith is the person who
doesn't ask Allah will be hated by Him, and the hated, is a person
Allah is angry at, and Allah loves/likes to be asked"
An other addition to this from Ibn Hajjar is that this Hadith (which is narrated by abu Huraira) is supported by two other ahadith one narrated by ibn Mas'ud (qualified as da'if) and one by Abdullah ibn 'Omar (qualified as da'if) (May Allah be pleased with them) which is also in al-Hakim's mustadrak with a "not strong sanad".
How to deal with a Hadith qualified as da'if?
Weak (Da'if) Hadith is only reliable in excellence, not in rulings (permissibility or prohibition will not be proven for it. Yes, good deeds or the prominence of an individual can be). (For details see here)
For this reason if a site like islamqa added that the Hadith is da'if then it's totally allowed to quote it and especially if the Hadith is not applied in things that are related to haram or halal rules like good deeds (some scholars add conditions like that the Hadith shouldn't have an extreme/high weakness).
Therefore one should understand this "Hadith" as an invitation to perform dua' as Allah likes to be asked!
And Allah knows best!