  Nuclei in the $Z\!\approx\!40,N\!\approx\!60$ region have one of the most
complicated structural evolution across the nuclear chart, with coexisting
shapes arising from different mixed configurations. In such a region, it is
difficult to investigate odd-mass nuclei. In this paper a new algebraic
framework is introduced, the interacting boson-fermion model with configuration
mixing. Using this framework, with a boson core and a proton in the
$1f_{5/2},2p_{3/2},2p_{1/2},1g_{9/2}$ orbits, a calculation is carried out to
understand the structural evolution of the odd-mass niobium isotopes $(Z=41)$
with neutron number 52-62. The calculated results are compared to energy
levels, two neutron separation energies, $E2$ and $M1$ transition rates, and to
quadrupole and magnetic moments. The detailed analysis discloses the effects of
an abrupt crossing of states between normal and intruder configurations (Type
II QPT), which is accompanied by a gradual evolution from spherical- to
deformed-core shapes within the intruder configuration (Type I QPT), where both
types of QPTs occur around the critical point of neutron number 60. The
identification of both types of QPTs in the same chain of isotopes provides an
empirical manifestation of intertwined quantum phase transitions (IQPTs) in
odd-mass nuclei and the relevance of IQPTs to the niobium chain.
