The change of variables $x=X/12$ and $y=Y/36$ gives the equation $$ Y^2 = X^3 - 18X^2 + 288X + + 1296. $$ Entering this into the LMFDB leads to the page http://www.lmfdb.org/EllipticCurve/Q/315936/g/1 . So your elliptic curve, after another change of variables to get rid of the $X^2$ term (replace $X$ by $X+6$) gives the curve $$ Y^2 = X^3 + 180X + 2592. \quad(*) $$ According to the LMFDB, this curve has Mordell-Weil rank 2, generated by $ \left(-6, 36\right) $ and $ \left(18, 108\right) $. And in the form $(*)$, there are the following integral points, where only the point with positive $Y$ is listed: $$ (-6, 36) ,\; (-3, 45) ,\; (18, 108) ,\; (24, 144) ,\; (28, 172) ,\; (738, 20052) ,\; (1722, 71460) ,\;(6189, 486891) .$$ This leads to the points $$ (0, 1),\; (1/4, 5/4),\; (2, 3),\; (5/2, 4),\; (17/6, 43/9),\; (62, 557),\; (144, 1985),\; (2065/4, 54099/4) $$ on your original curve. giving the 4 integral points that you found.

Start with $$ 3𝑦^2=4𝑥^3−6𝑥^2+8𝑥+3. $$ The change of variables $x=X/12$ and $y=Y/36$ gives the equation $$ Y^2 = X^3 - 18X^2 + 288X + 1296. $$ Entering this into the LMFDB leads to the page http://www.lmfdb.org/EllipticCurve/Q/315936/g/1 . So your elliptic curve, after another change of variables to get rid of the $X^2$ term (replace $X$ by $X+6$) gives the curve $$ Y^2 = X^3 + 180X + 2592. \quad(*) $$ According to the LMFDB, the curve $(*)$ has Mordell-Weil rank 2, generated by $ \left(-6, 36\right) $ and $ \left(18, 108\right) $. And in the form $(*)$, there are the following integral points, where only the point with positive $Y$ is listed: $$ (-6, 36) ,\; (-3, 45) ,\; (18, 108) ,\; (24, 144) ,\; (28, 172) ,\; (738, 20052) ,\; (1722, 71460) ,\;(6189, 486891) .$$ This leads to the points $$ (0, 1),\; (1/4, 5/4),\; (2, 3),\; (5/2, 4),\; (17/6, 43/9),\; (62, 557),\; (144, 1985),\; (2065/4, 54099/4) $$ on your original curve. giving the 4 integral points that you found.