We carried out an unbiased survey for massive dense cores in the giant molecular cloud associated with \eta Carinae with the NANTEN telescope in ^ { 12 } CO , ^ { 13 } CO , and C ^ { 18 } O J = 1–0 emission lines . We identified 15 C ^ { 18 } O cores , whose typical line width \Delta V _ { comp } , radius r , mass M , column density N ( H _ { 2 } ) , and average number density n ( H _ { 2 } ) were 3.3 km s ^ { -1 } , 2.2 pc , 2.6 \times 10 ^ { 3 } M _ { \sun } , 1.3 \times 10 ^ { 22 } cm ^ { -2 } , and 1.2 \times 10 ^ { 3 } cm ^ { -3 } , respectively . Two of the 15 cores are associated with IRAS point sources whose luminosities are larger than 10 ^ { 4 } L _ { \sun } , which indicates that massive star formation is occuring within these cores . Five cores including the two with IRAS sources are associated with MSX point sources . We detected H ^ { 13 } CO ^ { + } ( J = 1–0 ) emission toward 4 C ^ { 18 } O cores , two of which are associated with IRAS and MSX point sources , another one is associated only with an MSX point source , and the other is associated with neither IRAS nor MSX point sources . The core with neither IRAS nor MSX point sources shows the presence of a bipolar molecular outflow in ^ { 12 } CO ( J = 2–1 ) , which indicates that star formation is also occuring in the core , and the other three of the four H ^ { 13 } CO ^ { + } detections show wing-like emission . In total , six C ^ { 18 } O cores out of 15 ( = 40 \% ) are experienced star formation , and at least 2 of 15 ( = 13 \% ) are massive-star forming cores in the \eta Car GMC . We found that massive star formation occurs preferentially in cores with larger N ( H _ { 2 } ) , M , n ( H _ { 2 } ) , and smaller ratio of M _ { vir } / M . We also found that the cores in the \eta Car GMC are characterized by large \Delta V and M _ { vir } / M on average compared to the cores in other GMCs observed with the same telescope . These properties of the cores may account for the fact that as much as 60–87 % of the cores do not show any signs of massive star formation . We investigated the origin of a large amount of turbulence in the \eta Car GMC . We found that turbulence injection from stellar winds , molecular outflows , and supernova remnants which originated from stars formed within the GMC , are not enough to explain the existing turbulence . We propose the possibility that the large turbulence was pre-existing when the GMC was formed , and is now dissipating . Mechanisms such as multiple supernova explosions in the Carina flare supershell may have contributed to form a GMC with a large amount of turbulence .