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Here is one tedious way, where we have to homogenize to use the letterplace implementation:
sage: F.<Y,YINV,ONE1,ONE2> = FreeAlgebra(QQ, implementation='letterplace', degrees=[1,1,1,2])
sage: R.<y,yinv,one1,one2> = F.quotient(F*[Y*YINV - ONE2, YINV*Y - ONE2, ONE1*ONE1 - ONE2, Y*ONE1 - ONE1*Y, ONE1*YINV - YINV*ONE1]*F)
sage: y*(y+one1)*yinv
one2*y + one2*one1
If you squint, this is y + 1.
