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We consider the concept of fuzzy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>-quasi-contraction (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mi mathvariant="script">H</mi></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">Q</mi><mi mathvariant="script">C</mi></mrow></semantics></math></inline-formula> for short) initiated by Ćirić in tripled fuzzy metric spaces (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">T</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mi mathvariant="script">M</mi><mi mathvariant="script">S</mi></mrow></semantics></math></inline-formula>s for short) and present a new fixed point theorem (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mi mathvariant="script">P</mi><mi mathvariant="script">T</mi></mrow></semantics></math></inline-formula> for short) for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mi mathvariant="script">H</mi></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">Q</mi><mi mathvariant="script">C</mi></mrow></semantics></math></inline-formula> in complete <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">T</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mi mathvariant="script">M</mi><mi mathvariant="script">S</mi></mrow></semantics></math></inline-formula>s. As an application, we prove the corresponding results of the previous literature in setting fuzzy metric spaces (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mi mathvariant="script">M</mi><mi mathvariant="script">S</mi></mrow></semantics></math></inline-formula>s for short). Moreover, we obtain theorems of sufficient and necessary conditions which can be used to demonstrate the existence of fixed points. In addition, we construct relevant examples to illustrate the corresponding results. Finally, we show the existence and uniqueness of solutions for integral equations by applying our new results.