There is a website, Earliest Known Uses of Some of the Words of Mathematics. Some entries are,
DIOPHANTINE ANALYSIS (named for Diophantus of Alexandria) occurs in French in a letter of March 1770 from Euler to Lagrange: “ce problème me paraissait d'une nature singulière et surpassait même les règles connues de l'analyse de Diophante” (“this problem appeared to me to be of a singular nature and surpassed the known rules of Diophantine analysis”).
Lagrange used “analyse de Diophante” in a letter to D’Alembert in June 1771.
“Diophantine Analysis” occurs in English in the chapter title “Demonstration of a Theorem in the Diophantine Analysis. By Mr. P. Barlow, of the Royal Military Academy, Woolwich.” in The Mathematical Repository, New Series, Volume III (1809) page 70.
[This entry was contributed by James A. Landau.]
DIOPHANTINE EQUATION. Felix Klein used Diophantische Gleichungen in “Die Eindeutigen automorphen Formen vom Geschlechte Null” in the 1892 issue of Nachrichten (page 286): “Die Relationen kann man in Diophantische Gleichungen umsetzen, welche dann leicht übersehen lassen, unter welchen Umständen Multiplicatorsysteme möglich sind, und in welcher Anzahl.” [James A. Landau]
Diophantine equation appears in English in 1893 in Eliakim Hastings Moore (1862-1932), "A Doubly-Infinite System of Simple Groups," Bulletin of the New York Mathematical Society, vol. III, pp. 73-78, October 13, 1893 [Julio González Cabillón].
Henry B. Fine writes in The Number System of Algebra (1902):
The designation "Diophantine equations," commonly applied to indeterminate equations of the first degree when investigated for integral solutions, is a striking misnomer. Diophantus nowhere considers such equations, and, on the other hand, allows fractional solutions of indeterminate equations of the second degree. DIOPHANTINE PROBLEM. The phrase "Diophantus Problemes" appears in 1670 [James A. Landau].
The OED2 has the citation in 1700: Gregory, Collect. (Oxf. Hist. Soc.) I. 321: "The resolution of the indetermined arithmetical or Diophantine problems."