The ATLAS detector at CERN's LHC is preparing to take data from the
first proton-proton collisions expected in the next few months. We
report on the analysis of simulated data samples for production of heavy
Quarqonium states J/ψ→μμ and Υ→μμ, corresponding to an integrated
luminosity of 10 pb-1 with center of mass energy of 14 TeV expected at
the early ATLAS data. The report will review various aspects of prompt
Quarqonium production at LHC: the accessible ranges in transverse
momentum and pseudorapidity, spin alignment of vector states, separation
of color octet and color singlet production mechanism and feasibility of
observing radiative decays of χ c   and χ b  decays. Strategies of
various measurements are outlined and methods of separating promptly
produced J/ψ and Υ mesons from various backgrounds are discussed.