Choose an intersection as the point of origin.
The rules for areas of effect on a grid require that you choose an intersection of squares or hexes as the point of origin. From the Dungeon Master’s Guide (pp. 250-251), in the rules for using miniatures, it says the following about areas of effect:
Choose an intersection of squares or hexes as the point of origin of an area of effect, then follow its rules as normal. If an area of effect is circular and covers at least half a square, it affects that square.
So you would choose one of the corners of the space you are occupying as the point of origin for gust of wind.
Xanathar’s Guide to Everything (pp. 86-88) has some additional guidance in the section "Areas of Effect on a Grid", and even includes examples of how to adjudicate a 10-foot wide line on a grid: