Close hyperbolic encounters in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math>

We explore the dynamics and gravitational-wave emission from black hole pairs on unbound orbits undergoing close hyperbolic encounters (CHEs) in dense astrophysical environments. While general relativity predicts gravitational bremsstrahlung radiation occurring at periastron, the contribution of the scalar gravitational-wave mode in fully general $f(R)$ gravity remains largely unexplored. We characterize the $f(R)$ scalar mode gravitational radiation for both nonprecessing and precessing hyperbolic orbits, identifying potential detection signatures for advanced gravitational-wave observatories. By systematically varying orbital precession and eccentricity, we examine their influence on the emitted gravitational waves. We derive potentially detectable time delay and scalar-to-tensor amplitude ratio estimates for representative astrophysical environments and determine optimal orbital configurations for the detection of $f(R)$ scalar gravitational waves from hyperbolic encounters. Our results provide the theoretical framework for scalar mode signals and observables, establishing CHEs as a promising probe of $f(R)$ gravity with future detectors.