Hello
First thing - does the bike already have a (fixed length) quill stem? If it has a stem that clamps around the outside of the fork's steerer tube (a 'threadless stem') then you won't be able to fit a quill.
Your best bet is probably to have a look at this site:
https://www.sheldonbrown.com/gloss_st-z.html#stem. Also worth bookmarking that site as it contains a wealth of data and advice that could be useful when maintaining or modifying bikes.
I would have expected your 'steerer outside diameter' to be 1" meaning the 'wedge bolt" (quill) stem size would be 22.2mm, not the 20mm you have measured. If you're sure about that maybe someone else can advise whether smaller sizes appear on children's bikes? The other critical dimension is the attachment to the handlebars - the outside diameter of the bars where they are clamped by the stem is most likely to be 25.4mm