For those not watching the series I'm assuming that G1 is the baby/colt/whatever? So the ones arguing got the other pregnant?
i think G1 refers to Generation 1? in terms of breeding or something like that
G1 refers to Grade I stakes. Which is the highest level of competition for horse racing. When you win that competition you get a lot of prestige and money. I believe Grass-chan is saying Spe-chan and her should be tied for first place by becoming one. Thus becoming the most renowned horse racing couple.
In obstetrics, we refer pregnancy as "Gestation". G1 means first pregnancy, G2 means second pregnancy, and so on.