Let's take the I-IV-V-I progression for example. Can the I-IV-V-I progression in a minor key, after some transposing, sound like the I-IV-V-I in a major key?
No when you transpose chords the quality of the chord stays the same, but the notes making up the chord change.
For example, if you had the progression Am - Dm - Em - Am (i - iv - v - i in A minor) and tried to transpose it up a minor third to C, you would just have the progression Cm - Fm - Gm - Cm (i - iv - v - i in C minor). You would need to alter the qualities of the chords to get it to be C - F - G - C (I - IV - V - I in C major).
If you were however to change the key signature of the piece from the minor key to the major key this will turn the progression in the example from to A - D - E - A (I - IV - V - I in A major) assuming all the notes you were using were natural to the key. This is not a typical transposition and more of a change of notes based on a change of key signature and the modal change user25450 is also a valid way to look at this.
A proper transposition will never change the mode of a piece. However, there is such a thing as a "modal transposition" where you change in scale (rather than move the whole scale to a different pitch), adjusting the note steps rather than their chromatic pitches while retaining the current scale. So a modal transposition from C to A will also change C major to A minor. Stuff becomes more hand-wavy when you don't use natural minor but rather harmonic/melodic minor. You'll still retain the step distances, but won't necessarily get away without changing accidentals in addition.
Of course, one can combine chromatic transposition with modal transposition in order to go from C major to D minor (for example).
Transposing is by interval, and is usually applied to an instrument, or a entire song. So the short answer is no, transposing a chord progression will never change the quality of the music (here quality means major, minor, etc).
For example, if you played an instrument that, when you 'see' a written C note your instrument sounds a Bb, then your sheet music would need to be transposed to accommodate that.
Transposing is also commonly done for voice or melody instruments. In that case we would transpose the song to a key that better suits the voice or melody instrument.
What your are asking doesn't make sense because you are (probably) misusing the word transpose.
The closest things I can think of that might answer your question:
1) Re-Harmonization: Take the same melody, or with little to no change to the melody and use different chords.
2) Mode Change: The best examples are in classical music. This is when we suddenly change from major to minor, minor to major, etc.
3) Certain sections of a sonata will often have themes or subjects in different keys. Different subject, so I won't dive into this.
3) Relative and Distant Modulation, Mode Mixture, Borrowed and Altered Chords: Too much to explain here, but a brief would be that Modulation is a brief change in tonal center. Borrowed/Altered Chords, and Mode Mixture often are an effect of modulation.