No. The COP31 visa exists only for people the UNFCCC has approved as participants. Anyone travelling with you who is not registered has to use the ordinary Turkish visa route, if their passport needs one.
You also cannot apply on their behalf. The presidency is explicit that accompanying persons who are not registered participants are not eligible and must follow the standard Turkish visa application procedures where those apply.
In practice this is a smaller obstacle than it sounds, because Türkiye’s ordinary e-visa is open to many nationalities and is applied for on the same platform. It is worth separating the two applications in your head early, though, since the free COP31 route and its seven-day processing do not extend to anyone outside the badge.