No, this won't work for several reasons
There are several problems here...
Qualith is not the written form of Deep Speech. Deep Speech does not have a written form. It is the only language listed in the PHB that lacks a script used to write it down (see PHB/Basic Rules Chapter 4). Qualith is a unique, written-only language only used or understood by Mind Flayers. (MM, p. 221) There is nothing in any book that I can find that links it to Deep Speech.
On the rare occasion that mind flayers need to write something down, they do so in Qualith. This system of tactile writing (similar to braille) is read by an illithid’s tentacles. Qualith is written in four-line stanzas and is so alien in construction that non-illithids must resort to magic to discern its meaning. Though Qualith can be used to keep records, illithids most often use it to mark portals or other surfaces with warnings or instructions.
Secondly, while a Changeling can take an Illithid's form, they do not acquire the mind of an Illithid. As stated in the above linked text:
non-illithids must resort to magic to discern its meaning.
A Changeling's Change Form states...
none of your game statistics change.
(Changelings, WGtE, p. 61)
The languages you speak is a 'game statistic.' Thus, shapeshifting does not grant you new languages. You can't shapeshift to look like a dragonborn and auto-magically learn Draconic.
So, you don't get an Illithid's brain out of the deal, and Qualith is a unique written language only understood by Illithids because its construction is so alien that it makes no sense to anyone else. You could, theoretically, use your face tentacles to touch the tactile bits that make up Qualith, but you won't understand it any more than if you did the same thing with your fingers.