Do ionic bonds occur between atoms from adjacent groups?
No. Only metals and non-metals.
Ionic bonds only occur between a metal (a cation) and a non-metal (an anion).
This is because of electrostatic attraction between two oppositely charged ions.
You asked if they can form from atoms of adjacent groups.
This is not true. The reason why metals and non-metals bond due to electrostatic attraction, is because metals tend to lose electrons, while non-metals tend to gain electrons.
HOWEVER, metalloids cannot form ionic compounds with metals.
In this GIF, sodium (the metal) donates its electron to fluorine (the non-metal). As a result, they both become stable.
This relationship is so effective in achieving atomic stability that because of this, it is difficult to separate the atoms. It is also difficult to change the state of the compound.
Hope this helps :)