When we are on holiday can we pray Qasr (shortening the prayer) although we have enough time and mosque is very close to the house where we stay? Also explain when can we pray Qasr or Jamm? Please give authentic Hadith or Qur'an references.

Praise be to Allah, the Lord of the World; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

As a rule, a traveller can shorten prayers (four Rak'as to two) and combine them (Zuhr with 'Asr and Maghrib with 'Isha) in the fixed time of the first prayer or the second prayer. One's travel must be for a permissible goal and for a distance of 83 Km. at least.
Thereupon, if someone travels to some country where he has no wife nor intends to stay there for more than four days, he can shorten and combine prayers even though a mosque is near him. But, if shortening prayers will cause him to perform them alone whereas he has access to a congregation, then one has to perform prayers in full with the congregation according to the preponderant opinion of the Muslim scholars.
In a sound Hadith: "We went out with Allah's Messenger (Sallallahu Alaihi wa Sallam) for Tabook Ghazwa (a military expedition led by Prophet Muhammad) and he performed Zuhr and 'Asr prayers combined, and Maghrib and 'Isha combined". [Reported by Imam Muslim]
Allah knows best.